27993 - Mathematical Analysis T-2

Course contents

THE EUCLIDEAN SPACE R^n. The vector space structure, the dot product and the euclidean norm. Open, closed, bounded, compact, connected subsets of R^n.

LIMITS, CONTINUITY AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES.

Generalities on real and vector functions of several real variables. Definition of limit of a function and continuous function and of . The Weierstrass, zeros and Heine-Cantor's theorem and the intermediate value theorem for functions of several variables. Partial and directional derivatives. Differentiable and C^1 functions; the differential and the Jacobian matrix. The chain rule. Partial derivatives of higher order. Hessian matrix. Taylor's formula of the second order for functions of several variables. Interior and constrained local extrema for real functions of several variables.

CURVE INTEGRALS.

Curves, length of a curve, orientation. Integral of a function over a curve.

The integral of a vector field over an oriented curve. Conservative vector fields and their potentials. Work of a vector field.

MULTIPLE INTEGRALS.

Normal domains. Double and triple integrals. The reduction formula. The change of variables theorem for a double integral.Gauss-Green's formulas and Stokes'Theorem in the plane.

SURFACE INTEGRALS.

Smooth surfaces. Tangent plane and normal vector. Area of a surface. Integral of a function over a surface. The divergence theorem and the Stokes theorem.

DIFFERENTIAL EQUATIONS. The Cauchy problem. Linear equations. Equations with separable variables. 

Readings/Bibliography

Fusco-Marcellini-Sbordone: Analisi Matematica Due, Liguori Editore.

Exercises:
Bramanti M.: Esercitazioni di Analisi Matematica 2 , Ed. Esculapio.

More reference texts:

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, ed. Zanichelli

V. Barutello, M. Conti, D. Ferrario, S. Terracini, G. Verzina: Analisi Matematica vol. 2, ed. Apogeo

M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica, seconda edizione (2011) Mc Graw Hill

Teaching methods

The course consists of lessons describing the fundamental concepts of functions of two or more real variables and differential equations. Lessons are completed with examples and counterexamples illuminating the theoretical content. Futhermore many exercises are solved in the classroom.

Assessment methods

The examination consists of a preliminary written test (exercises) and a test about the theoretical part.
The preliminary written test (exercises) lasts three hours and consists in solving about six exercises related to the arguments of the course. In order to sustain it the student must register at the test through AlmaEsami [https://almaesami.unibo.it/] . If this written test is passed (at least 15/30), the student can sit for the test concerning the theoretical aspects of the course. The student must show to know the concepts explained during the course (in particular definitions and theorems) and how to connect them. The theoretical part of the exam must be passed in the same "sessione" of the preliminary written test (exercises): in the same "appello" or in the subsequent one.

Teaching tools

Tutorship (if appointed)

Office hours

See the website of Giovanni Cupini